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Could do with a bit more info - do you mean to run a function or actually open an new application (.php page) window?

If you're asking for a php equivelant to javascript's onclick="DoFunction();" action, then no, there isnt one (php is server-side, javascript is client-side)

But if you're asking if you can insert HTML into a php file to display your logo image, then yes, just use the usual <a href="my_application.php"><img src="path/to/image/my_logo.jpg" height="100" width="100" alt="logo" /></a> code and wrap it in hyperlinked anchor tags to open your application.php page.

You will need a .ico file and a way to attach this to your program. The easiest way would be to use your compiler if there is an option, such as compile test.php -i:icon.ico (rough guess about the format), if your compiler supports that.
If your compiler does not, then you will need to find a way to change the icon of an .exe that is already compiled. I don't know if this is possible, but if it is I think you can find shareware/freeware for it. Try searching google for "set icon .exe" and have some patience-- you'll get a lot of useless results (mostly for creating .ico files-- but you might need that too), and probably some that work.

Note: this question is completely unrelated to PHP (as code), and only related to your compiler and how windows works with .exe files. You might find more help at a desktop programming forum if needed. Most users here don't use PHP to create desktop applications (and most people who DO create desktop applications don't use PHP). I've done a very small amount, but I don't know very much about it except that PHP is very slow and usually not the right language for desktop programming.

I don't know what compiler you are using, but there may be no reason not to use bambalam:http://www.bambalam.se/bamcompile/
It's free and supports icons (in the format I used above).
This is the compiler I've used for PHP. It works well, at least for basic programs.

the whole idea of desktop php is mildly exciting for me... my initial reaction is "wow, awesome, I'd love to try that!", but then my sense of practicality comes in: I don't have any needs for a desktop app, no experience in writing them at all (not since 5th grade when we did stuff like 10 IF X = 42 GOTO 40
20 X = X + 1
30 GOTO 10
40 PRINT "X equals 42", anyway), and, beyond that, I get the feeling that if I ever get into desktop programs, there's probably a better place to start than php.

I've used it and tried a few things. It works, but it's slow.
I was using the GD library at one point to do some filters for images, and I created an .exe that would run through all of the images in a folder and apply it to all of them then save the output files. GD is inefficient and running PHP as a desktop app is inefficient, but aside from that it was good and did what I needed. The main advantage of running it there instead of my server was that it could run for hours (literally) without the problems of doing that on a server. And I could use files locally rather than uploading them.
Other things I've tried are a file encryptor (just for fun but it worked) and an FTP uploader (using PHP's ftp functions). I wanted to get the ftp uploader to be more useful (as it was, I compiled the filename and destination into the program), but never got a chance.
The main problem is that the compiler is simple and doesn't have many options for input/output, etc. It actually does have these options, but it's complicated and would as you say be easier to do it using another language. Since I don't know another language, though, it's better than nothing and fun to play with.

As a completely random note, at least with bambalam, you can compile an include then keep the rest as text (included.php). So I found this a good way to test things (mostly image manipulations), and this might be a good way to test some complex algorithm locally rather than dealing with it on your server.